Chapter 8.5, Problem 60E

a.

To determine

To write: a recursive rule for the number of $a_n$ books in the library at the beginning of the $nth$ year.

Answer to Problem 60E

  $a_n=0.98a_{n-1}+1150$ .

Explanation of Solution

Given:

A town library initially has 54,000 books in its collection. Each year, 2% of the books are lost or discarded. The library can afford to purchase 1150 new books each year.

Calculation:

According to the question, 54,000 books in its collection. Each year, 2% of the books are lost or discarded. The library can afford to purchase 1150 new books each year.

  $\begin{array}{l}{a}_1=54,000\\ a_n=0.98a_{n-1}+1150\end{array}$ .

Now, write a recursive rule for the number of $a_n$ books:

  $\begin{array}{l}f\left(1\right)=54,000\\ f\left(2\right)=54,070\\ f\left(3\right)=54,138.6\\ f\left(4\right)=54,205.827999999994\\ f\left(5\right)=54,271.71143999999\\ f\left(6\right)=54,336.27721119999\\ f\left(7\right)=54,399.55166697599\\ f\left(8\right)=54,461.560633636465\\ f\left(9\right)=54,522.32942096373\\ f\left(10\right)=54,581.882832544456\\ f\left(11\right)=54,640.245175893564\\ f\left(12\right)=54,697.440272375694\end{array}$

  $\begin{array}{l}f\left(13\right)=54,753.49146692818\\ f\left(14\right)=54,808.421637589614\\ f\left(15\right)=54,862.25320483782\\ f\left(16\right)=54,915.00814074106\\ f\left(17\right)=54,966.70797792624\\ f\left(18\right)=55,017.37381836772\\ f\left(19\right)=55,067.02634200036\\ f\left(20\right)=55,115.68581516035\\ f\left(21\right)=55,163.372098857144\\ f\left(22\right)=55,210.10465688\\ f\left(23\right)=55,255.9025637424\\ f\left(24\right)=55,300.78451246755\end{array}$

Hence, a recursive rule for the amount of chlorine in the pool at the start of the mil week is $a_n=0.98a_{n-1}+1150$ .

b.

To determine

To find: the amount of chlorine in the pool at the start of the third week.

Answer to Problem 60E

  $a_3=37.84$ .

Explanation of Solution

Given:

A town library initially has 54,000 books in its collection. Each year, 2% of the books are lost or discarded. The library can afford to purchase 1150 new books each year.

Calculation:

From part (a) $a_n=0.8a_{n-1}+5000$ .

Therefore, start of the 3rd week is

  $\begin{array}{l}{a}_n=0.60a_{n-1}+16\\ a_3=0.6a_2+16=37.84\end{array}$

Hence, the number of members at the start of the 5th year is $a_3=37.84$ .

c.

To determine

To describe: what happens to the amount of chlorine in the pool over time.

Answer to Problem 60E

40.

Explanation of Solution

Given:

The initial amount of chlorine in a swimming pool is $34\text{ounces}$ .

And the amount of chlorine is continuously added 16 ounces and 40% of the chlorine evaporates.

Calculation:

From part (a)

  $\begin{array}{l}{a}_1=34\\ a_3=37.84\end{array}$ .

  $\begin{array}{l}{0.6}^{n-1}\left(34\right)\to 0\\ 16\cdot \frac{1-{0.6}^{n-1}}{1-0.6}\to 16\cdot \frac{1}{1-0.6}\\ =16\cdot \frac{1}{0.4}=16\times \frac{10}{4}=40\end{array}$

Hence, he sequence approaches to 40.